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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.sf_poly.legendre"></a><a class="link" href="legendre.html" title="Legendre (and Associated) Polynomials"> Legendre (and |
| Associated) Polynomials</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.sf_poly.legendre.synopsis"></a><h5> |
| <a name="id1110272"></a> |
| <a class="link" href="legendre.html#math_toolkit.special.sf_poly.legendre.synopsis">Synopsis</a> |
| </h5> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">legendre</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span> |
| |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result |
| type calculation rules</em></span></a>: note than when there is a single |
| template argument the result is the same type as that argument or <code class="computeroutput"><span class="keyword">double</span></code> if the template argument is an integer |
| type. |
| </p> |
| <p> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument |
| is optional and can be used to control the behaviour of the function: |
| how it handles errors, what level of precision to use etc. Refer to the |
| <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. |
| </p> |
| <p> |
| </p> |
| <a name="math_toolkit.special.sf_poly.legendre.description"></a><h5> |
| <a name="id1111412"></a> |
| <a class="link" href="legendre.html#math_toolkit.special.sf_poly.legendre.description">Description</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the Legendre Polynomial of the first kind: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/legendre_0.png"></span> |
| </p> |
| <p> |
| Requires -1 <= x <= 1, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </p> |
| <p> |
| Negative orders are handled via the reflection formula: |
| </p> |
| <p> |
| P<sub>-l-1</sub>(x) = P<sub>l</sub>(x) |
| </p> |
| <p> |
| The following graph illustrates the behaviour of the first few Legendre |
| Polynomials: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/legendre_p.png" align="middle"></span> |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_p</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the associated Legendre polynomial of the first kind: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/legendre_1.png"></span> |
| </p> |
| <p> |
| Requires -1 <= x <= 1, otherwise returns the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </p> |
| <p> |
| Negative values of <span class="emphasis"><em>l</em></span> and <span class="emphasis"><em>m</em></span> are |
| handled via the identity relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/legendre_3.png"></span> |
| </p> |
| <div class="caution"><table border="0" summary="Caution"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../doc/src/images/caution.png"></td> |
| <th align="left">Caution</th> |
| </tr> |
| <tr><td align="left" valign="top"> |
| <p> |
| The definition of the associated Legendre polynomial used here includes |
| a leading Condon-Shortley phase term of (-1)<sup>m</sup>. This matches the definition |
| given by Abramowitz and Stegun (8.6.6) and that used by <a href="http://mathworld.wolfram.com/LegendrePolynomial.html" target="_top">Mathworld</a> |
| and <a href="http://documents.wolfram.com/mathematica/functions/LegendreP" target="_top">Mathematica's |
| LegendreP function</a>. However, uses in the literature do not always |
| include this phase term, and strangely the specification for the associated |
| Legendre function in the C++ TR1 (assoc_legendre) also omits it, in spite |
| of stating that it uses Abramowitz and Stegun as the final arbiter on |
| these matters. |
| </p> |
| <p> |
| See: |
| </p> |
| <p> |
| <a href="http://mathworld.wolfram.com/LegendrePolynomial.html" target="_top">Weisstein, |
| Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web |
| Resource</a>. |
| </p> |
| <p> |
| Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions" |
| and "Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook |
| of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, |
| 9th printing. New York: Dover, pp. 331-339 and 771-802, 1972. |
| </p> |
| </td></tr> |
| </table></div> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_q</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the value of the Legendre polynomial that is the second solution |
| to the Legendre differential equation, for example: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/legendre_2.png"></span> |
| </p> |
| <p> |
| Requires -1 <= x <= 1, otherwise <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| is called. |
| </p> |
| <p> |
| The following graph illustrates the first few Legendre functions of the |
| second kind: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/legendre_q.png" align="middle"></span> |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span> |
| </pre> |
| <p> |
| Implements the three term recurrence relation for the Legendre polynomials, |
| this function can be used to create a sequence of values evaluated at the |
| same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>l</em></span>. This |
| recurrence relation holds for Legendre Polynomials of both the first and |
| second kinds. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/legendre_4.png"></span> |
| </p> |
| <p> |
| For example we could produce a vector of the first 10 polynomial values |
| using: |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// Abscissa value |
| </span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">;</span> |
| <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)).</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> |
| <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span> |
| <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_next</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span> |
| </pre> |
| <p> |
| Formally the arguments are: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl> |
| <dt><span class="term">l</span></dt> |
| <dd><p> |
| The degree of the last polynomial calculated. |
| </p></dd> |
| <dt><span class="term">x</span></dt> |
| <dd><p> |
| The abscissa value |
| </p></dd> |
| <dt><span class="term">Pl</span></dt> |
| <dd><p> |
| The value of the polynomial evaluated at degree <span class="emphasis"><em>l</em></span>. |
| </p></dd> |
| <dt><span class="term">Plm1</span></dt> |
| <dd><p> |
| The value of the polynomial evaluated at degree <span class="emphasis"><em>l-1</em></span>. |
| </p></dd> |
| </dl> |
| </div> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T3</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">legendre_next</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">Pl</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">Plm1</span><span class="special">);</span> |
| </pre> |
| <p> |
| Implements the three term recurrence relation for the Associated Legendre |
| polynomials, this function can be used to create a sequence of values evaluated |
| at the same <span class="emphasis"><em>x</em></span>, and for rising <span class="emphasis"><em>l</em></span>. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/legendre_5.png"></span> |
| </p> |
| <p> |
| For example we could produce a vector of the first m+10 polynomial values |
| using: |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span> <span class="comment">// Abscissa value |
| </span><span class="keyword">int</span> <span class="identifier">m</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span> <span class="comment">// order |
| </span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">;</span> |
| <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="identifier">m</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)).</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_p</span><span class="special">(</span><span class="number">1</span> <span class="special">+</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">));</span> |
| <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span> <span class="special">+</span> <span class="identifier">m</span><span class="special">;</span> <span class="identifier">l</span> <span class="special"><</span> <span class="identifier">m</span> <span class="special">+</span> <span class="number">10</span><span class="special">;</span> <span class="special">++</span><span class="identifier">l</span><span class="special">)</span> |
| <span class="identifier">v</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="identifier">legendre_next</span><span class="special">(</span><span class="identifier">l</span><span class="special">,</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">],</span> <span class="identifier">v</span><span class="special">[</span><span class="identifier">l</span><span class="special">-</span><span class="number">1</span><span class="special">]));</span> |
| </pre> |
| <p> |
| Formally the arguments are: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl> |
| <dt><span class="term">l</span></dt> |
| <dd><p> |
| The degree of the last polynomial calculated. |
| </p></dd> |
| <dt><span class="term">m</span></dt> |
| <dd><p> |
| The order of the Associated Polynomial. |
| </p></dd> |
| <dt><span class="term">x</span></dt> |
| <dd><p> |
| The abscissa value |
| </p></dd> |
| <dt><span class="term">Pl</span></dt> |
| <dd><p> |
| The value of the polynomial evaluated at degree <span class="emphasis"><em>l</em></span>. |
| </p></dd> |
| <dt><span class="term">Plm1</span></dt> |
| <dd><p> |
| The value of the polynomial evaluated at degree <span class="emphasis"><em>l-1</em></span>. |
| </p></dd> |
| </dl> |
| </div> |
| <a name="math_toolkit.special.sf_poly.legendre.accuracy"></a><h5> |
| <a name="id1113804"></a> |
| <a class="link" href="legendre.html#math_toolkit.special.sf_poly.legendre.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following table shows peak errors (in units of epsilon) for various |
| domains of input arguments. Note that only results for the widest floating |
| point type on the system are given as narrower types have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively |
| zero error</a>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.special.sf_poly.legendre.peak_errors_in_the_legendre_p_function"></a><p class="title"><b>Table 29. Peak Errors In the Legendre P Function</b></p> |
| <div class="table-contents"><table class="table" summary="Peak Errors In the Legendre P Function"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| Errors in range |
| </p> |
| <p> |
| 0 < l < 20 |
| </p> |
| </th> |
| <th> |
| <p> |
| Errors in range |
| </p> |
| <p> |
| 20 < l < 120 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=211 Mean=20 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=300 Mean=33 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| SUSE Linux IA32, g++ 4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=70 Mean=10 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=700 Mean=60 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux IA64, g++ 3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=70 Mean=10 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=700 Mean=60 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=35 Mean=6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=292 Mean=41 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.sf_poly.legendre.peak_errors_in_the_associated_legendre_p_function"></a><p class="title"><b>Table 30. Peak Errors In the Associated Legendre P Function</b></p> |
| <div class="table-contents"><table class="table" summary="Peak Errors In the Associated Legendre P Function"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| Errors in range |
| </p> |
| <p> |
| 0 < l < 20 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1200 Mean=7 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| SUSE Linux IA32, g++ 4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=80 Mean=5 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux IA64, g++ 3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=80 Mean=5 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=42 Mean=4 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.sf_poly.legendre.peak_errors_in_the_legendre_q_function"></a><p class="title"><b>Table 31. Peak Errors In the Legendre Q Function</b></p> |
| <div class="table-contents"><table class="table" summary="Peak Errors In the Legendre Q Function"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| Errors in range |
| </p> |
| <p> |
| 0 < l < 20 |
| </p> |
| </th> |
| <th> |
| <p> |
| Errors in range |
| </p> |
| <p> |
| 20 < l < 120 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=50 Mean=7 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4600 Mean=370 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| SUSE Linux IA32, g++ 4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=51 Mean=8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=6000 Mean=480 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux IA64, g++ 3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=51 Mean=8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=6000 Mean=480 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=90 Mean=10 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1700 Mean=140 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><p> |
| Note that the worst errors occur when the order increases, values greater |
| than ~120 are very unlikely to produce sensible results, especially in |
| the associated polynomial case when the degree is also large. Further the |
| relative errors are likely to grow arbitrarily large when the function |
| is very close to a root. |
| </p> |
| <p> |
| No comparisons to other libraries are shown here: there appears to be only |
| one viable implementation method for these functions, the comparisons to |
| other libraries that have been run show identical error rates to those |
| given here. |
| </p> |
| <a name="math_toolkit.special.sf_poly.legendre.testing"></a><h5> |
| <a name="id1114392"></a> |
| <a class="link" href="legendre.html#math_toolkit.special.sf_poly.legendre.testing">Testing</a> |
| </h5> |
| <p> |
| A mixture of spot tests of values calculated using functions.wolfram.com, |
| and randomly generated test data are used: the test data was computed using |
| <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000-bit |
| precision. |
| </p> |
| <a name="math_toolkit.special.sf_poly.legendre.implementation"></a><h5> |
| <a name="id1114414"></a> |
| <a class="link" href="legendre.html#math_toolkit.special.sf_poly.legendre.implementation">Implementation</a> |
| </h5> |
| <p> |
| These functions are implemented using the stable three term recurrence |
| relations. These relations guarentee low absolute error but cannot guarentee |
| low relative error near one of the roots of the polynomials. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
| </div></td> |
| </tr></table> |
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